466
N. C. Bacalis
Keywords Configuration interaction · Excited states · Saddle point by
minimization · Truncated wave function · LUMO
13.1 Introductory Remark
Suppose that we want to know the behavior of some material when we bring
atoms close together, for example, we want to know the behavior of a catalyst or
we want to invent a new battery. After the contact of the atoms, where will the
electrons go? Will they be excited? How will they drive the surrounded nuclei?
Will they go to the lowest unoccupied orbital of the ground state or to the highest
occupied orbital of the excited state? When the charge transfer occurs in appropriate
excited electronic states, the electrons are transferred to appropriate orbitals of the
surrounding species. Therefore, we must know not only the excitation energy but
also the wave function of the excited state. But, the excited states are saddle
points in the Hilbert space, and the standard computational methods (except for
some that aim to the saddle nature of the answer), based on the Hylleraas-Undheim
and MacDonald (HUM) theorem [1], minimize the energy of the desired root of the
secular equation, since in optimizing the “desired root,” the other roots, all mutually
orthogonal, get deteriorated. However, a local minimum (i.e., HUM) cannot
be simultaneously saddle point in the same restricted subspace; therefore, the
standard methods, finding a local energy minimum, do not find the saddle point
sought, with a danger, if they use small truncated, supposedly easily comprehensible
expansions, to get probably a deceiving answer. For this reason the standard methods
resort to approaching the saddle point using huge expansions (rather inappropriate
for large systems). We can use a proposed functional [2] that, even within small
expansions, has minimum at the saddle point, successfully tested, up to now, by
direct multidimensional minimization, which is hard to use and time-consuming.
It will be much more beneficial to the scientific community, both theoretical and
practical, if we try to transform it, so that it is solved, faster and safer, selfconsistently, as we shall see below, but this has not been done yet. Presently we shall
see the presentation of the functional along with some computational applications
due to it, e.g., (i) immediately improving a ground state approximant if we know a
better excited state saddle point approximant and (ii) avoiding “root flipping.” Then,
we shall see demonstrations, for excited He, using both Hylleraas coordinates and
configuration interaction (CI) in standard coordinates, and for Li using CI.
13.2 Overview
In order to study electronic processes occurring via excited states, we must know
not only the excitation energy but also the wave function of the excited state, which,
as is well known, is a saddle point in the Hilbert space. The correct wave function
N. C. Bacalis
Keywords Configuration interaction · Excited states · Saddle point by
minimization · Truncated wave function · LUMO
13.1 Introductory Remark
Suppose that we want to know the behavior of some material when we bring
atoms close together, for example, we want to know the behavior of a catalyst or
we want to invent a new battery. After the contact of the atoms, where will the
electrons go? Will they be excited? How will they drive the surrounded nuclei?
Will they go to the lowest unoccupied orbital of the ground state or to the highest
occupied orbital of the excited state? When the charge transfer occurs in appropriate
excited electronic states, the electrons are transferred to appropriate orbitals of the
surrounding species. Therefore, we must know not only the excitation energy but
also the wave function of the excited state. But, the excited states are saddle
points in the Hilbert space, and the standard computational methods (except for
some that aim to the saddle nature of the answer), based on the Hylleraas-Undheim
and MacDonald (HUM) theorem [1], minimize the energy of the desired root of the
secular equation, since in optimizing the “desired root,” the other roots, all mutually
orthogonal, get deteriorated. However, a local minimum (i.e., HUM) cannot
be simultaneously saddle point in the same restricted subspace; therefore, the
standard methods, finding a local energy minimum, do not find the saddle point
sought, with a danger, if they use small truncated, supposedly easily comprehensible
expansions, to get probably a deceiving answer. For this reason the standard methods
resort to approaching the saddle point using huge expansions (rather inappropriate
for large systems). We can use a proposed functional [2] that, even within small
expansions, has minimum at the saddle point, successfully tested, up to now, by
direct multidimensional minimization, which is hard to use and time-consuming.
It will be much more beneficial to the scientific community, both theoretical and
practical, if we try to transform it, so that it is solved, faster and safer, selfconsistently, as we shall see below, but this has not been done yet. Presently we shall
see the presentation of the functional along with some computational applications
due to it, e.g., (i) immediately improving a ground state approximant if we know a
better excited state saddle point approximant and (ii) avoiding “root flipping.” Then,
we shall see demonstrations, for excited He, using both Hylleraas coordinates and
configuration interaction (CI) in standard coordinates, and for Li using CI.
13.2 Overview
In order to study electronic processes occurring via excited states, we must know
not only the excitation energy but also the wave function of the excited state, which,
as is well known, is a saddle point in the Hilbert space. The correct wave function